Algorithmic
Study of Infinite Simplicial Complexes

We study
different classes of constructive infinite simplicial complexes,
from an algorithmic point of vue. Firstly, we study the triangulated
non-compact
surfaces without boundary whose triangulations are defined by some
hyperedge
replacement graph grammars: the HR-equational surfaces. We give an
algorithm
to decide whether two such surfaces are homeomorphic or not.

Then, we
look at the case of bordered non-compact surfaces. To this
respect, we generalize the classification theorem of
Kerékjarto-Richards
to the case of planar non-compact surfaces.

In the
second part, we study the case of 3-dimensional manifolds.
We show that HR-equational non-compact hyperbolic 3-manifolds are
essentially
characterized by their fundamental groups. We then study these groups
which
are amalgamated products of infinitely many groups, but in finite
number
up to isomorphism. In particular, we define for them some constructive
presentations given in terms of rational languages.

The last
part of our work is devoted to automatic graphs which generalize
HR-equational graphs. To this respect, we show that the problem of
deciding
whether an automatic graph has more than one end is undecidable. On the
other hand, we show that this problem is equivalent to the one of
determining
whether all the graphs generated by a given graph D0L-system are
connected.
It turns out that this last question is also undecidable.